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\title{\vglue -20pt About Riemann's Minimal Surfaces}
\author{H. Karcher}
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   This is the family of singly-periodic embedded minimal
surfaces found by Riemann. They are parametrized (aa) by
rectangular tori. The Gauss map is the Weierstrass $\wp$
function additively normalized to have a double zero at
the branch point diagonally opposite the double pole and
multiplicatively normalized to have the values plus or
minus $i$  at the four midpoints (on the torus) between the
zero and the pole. The minimal surface has rotational
symmetries around the corresponding normals. This
symmetry kills the horizontal periods.  The surface is
parametrized by the range of the Gauss map with polar
coordinates around the punctures.
The surfaces look like families of parallel planes with one handle
between adjacent planes. The associate family morphing joins
two such embedded surfaces---they are congruent for the square
torus. The standard morphing (aa) changes the branch values of
the Gauss map, i.e. the tilt of the normal at the flat points ($K=0$).

  For a discussion of techniques for creating minimal surfaces with
various qualitative features by appropriate choices of Weierstrass
data, see either [KWH], or pages 192--217 of [DHKW].

[KWH]  H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and
         the minimal surfaces that led to its discovery, in ``Global Analysis
         in Modern Mathematics, A Symposium in Honor of Richard Palais'
         Sixtieth Birthday'', K. Uhlenbeck Editor, Publish or Perish Press, 1993

[DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab,
           Minimal Surfaces I, Grundlehren der math. Wiss. v. 295
           Springer-Verlag, 1991





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